If you are the only named beneficiary, the inherited 401(k) is entirely yours and you are responsible for the taxes on any distributions from the inherited 401(k). Any amounts of a distribution from this inherited 401(k) that you give to your siblings are gifts from you to your siblings. If you gift more than $14,000 to any one individual during a taxable year you are required to file Form 709 to report the gift.
Since you are paying the taxes on this income, It would make sense to gift to your siblings a proportional amount net of taxes that you must pay. For example, if you obtain a distribution of $16,000 and your marginal tax rate is 25%, that would leave $12,000 after taxes that you then gift as $3,000 to each of your three siblings.
To minimize taxes, you don't have to take a lump-sum distribution from the inherited 401(k). You can instruct the plan administrator to perform a direct rollover of the inherited 401(k) to an inherited IRA for your benefit, then make distributions over several years. You'll be subject to required minimum distributions based on your age. If your mother died after her required beginning date for RMDs, you'll have to complete the distribution of any unsatisfied portion of her year-of-death RMD before directly rolling over the remainder of the inherited 401(k).
Be aware that the rollover to an inherited IRA must be done by direct rollover. If any distribution is paid to you personally, it is not eligible for rollover. The plan administrator is required by law to provide you with details of your distribution options prior to making any distribution to you.
