How to memorize multiplication tables using a structured drill
This method of drilling is NOT initially random, but structured. This means that the student sees the structure of the particular multiplication table written in front of him, just without the answers. The teacher will point to the problems in a random fashion. Then the table is also practiced "backwards", again, with the complete table written in the correct order in front of the student, but the first multiplicands missing. Random drilling with flashcards or games is saved for later, after the student has mastered the table using this structured drill.
Seeing the structure of the table makes the task somewhat easier, because it reminds the children of the skipcounting pattern. The way this is done provides a visual, auditory, and kinesthetic cue for the student. The teacher points to various problems, MOVING his/her hand up/down the table, and the student can do the same movement. The teacher both POINTS to the problem (visual) and SAYS it out loud (auditory).
Please see the video below to fully understand the method. The goal of the drilling is to memorize a particular multiplication table, and the drilling should only be done after the child understands the CONCEPT of multiplication.
When you are doing drills to memorize, explain to the child that the goal is to memorize the facts, to recall from memory, and not to get the answers by counting or some other method. Just like your child probably has memorized your address and phone number, now she/he is going to memorize some math facts. You can easily see if the student is trying to count because producing the answer takes much more time. You should expect the answers from the child immediately when you are drilling. If he/she doesn't know the answer by heart (from memory), then tell him/her the right answer.
Usually short drill sessions are best. You can drill for 510 minutes at a time, depending on the child.
Try to have at least two sessions within a day though, as your schedule permits. Brain research shows us that forgetting happens fast, and that new information is retained far better if the first reviewing session is done within 46 hours of the first time learning. (This principle applies to anything new you are learning.)
Paperpencil activities where the child is left alone, do not work really well for memorizing the facts  the child may get the answers by counting and not from memory. So it will take time from the teacher/parent. If you can, utilize older siblings in the drilling task too. Computers are great drillers since they won't get tired and you can usually choose a timed session where the child is then forced to produce the answers quickly. Children can actually enjoy the memorization process when they notice they are truly learning the facts and are able to go through the drills successfully. Computer programs and computerbased drilling can be very rewarding to children and let them enjoy memorizing times tables. See a list of free online multiplication activities.
The method below has several steps from 1 to 5. You can work on only a few of the steps in one session, again, depending on the child's concentration and ability.
Memorizing the table of 3  in steps
Have a table to be worked on all ready written on paper. We will use here the table of three as an example.
2 × 3 = 6
3 × 3 = 9
4 × 3 = 12
5 × 3 = 15
6 × 3 = 18
7 × 3 = 21
8 × 3 = 24
9 × 3 = 27
10 × 3 = 30
11 × 3 = 33
12 × 3 = 36

The first task is to memorize the list of answers, so to speak.
Study first the skipcounting list up until the midpoint (3, 6, 9, 12, 15, 18).
Have your child say it alound while pointing to the answers one by one with
a finger or pen  thereby using many of his senses simultaneously. After he
has gone through if a few times, ask him to repeat the list from memory.
Try require the answers from your child, and not give them to her too easily, because ONLY by straining her mind will she make the effort to eventually memorize these facts. The mind is like muscles: it needs exercise to become stronger.
Require her to memorize this list both upwards and downwards. Continue this way until she can 'rattle off' the first list of 3, 6, 9, 12, 15, 18.
With some tables, like table of 2, table of 5, or table of 10, point out the pattern in them. The pattern in table of 9 is more subtle but still usable.
 Then tackle the last part of the list: 21, 24, 27, 30, 33, 36. Do the same things you did with the first part of the
list.
 Lastly, work with the whole list of answers. Practice the list UP AND DOWN until it goes smooth and easy.
This part may be enough for one day. But review it later in the day.

Next, practice
individual problems randomly. You can ask orally ("What is 5 times
3?") or point to the problems
on the paper, or use flashcards. However, I would recommend saying a
question aloud and simultaneously pointing to the problem the child can see,
because again, using multiple senses should help fix them in the mind better.
The goal at this stage is to associate each answer 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, with a certain multiplication fact (such as 7 × 3).
You can also mix earlier tables that she already knows with these new problems, and drill both with flashcards.

The last step is to do this the other way round so that YOU say the answer, say 21, and the
student has to produce the problem (from table of 3). Have the table handy, hide the problems, and point to the answers in random order.
This one you can work the other way around: the student says answers, and you produce problems. Answer wrong sometimes, too, to check her out.
As an extension, you can say answers from several tables that you've studied, and the student gives the corresponding problem. Sometimes there are several answers: for example 36, 30, 24, and 20 are in several different times tables. This is an especially good exercise as it prepares to division concept and factoring.
The
memorization won't probably happen overnight. On subsequent days, you
can mix these drills 15 (and hopefully you don't need to concentrate on
steps 1 and 2). This kind of drilling takes a little time and effort from the teacher, but it
can be very effective. And, homeschoolers can obviously do some of it
while going about other tasks, or while traveling in the car, etc.
While you are doing this table by table, you can also try to teach the process to your child, so that she will learn how to do memorization herself. She can hide the answers and try to produce the list in her mind.
Other helpful ideas

Hang a poster with the 12x12 or 10x10 grid on the wall. Remind your child to glance at it a few times a day. It can work wonders for visual learners!

Hang beside it another, initially empty, poster, to which the child fills in those facts he has mastered.

Recite the skipcounting lists or multiplication facts aloud just before going to bed. This can turn them into mastered facts by the next morning.