您可以删除以下第一个选项，无论其包含什么值。

$("#q_c_0_a_0_name").find("option").eq(0).remove();

<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
<select id="q_c_0_a_0_name">
 <option value="name">     Filter by Name</option>
 <option value="lastanem1">Filter by First Lastname</option>
 <option value="lastname2">Filter by Second Lastname</option>
 ...
 <option value="xxxxxxxxx">Filter by Other value</option>
</select>

使用:first / :first-child

$("#q_c_0_a_0_name option:first").remove();

<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>

<select id="q_c_0_a_0_name">
 <option value></option>
 <option value="lastanem1">First Lastname</option>
 <option value="lastname2">Second Lastname</option>
 <option value="xxxxxxxxx">Other value</option>
</select>

尝试此代码

$("#q_c_0_a_0_name option[value='name']").remove();

如果要删除第一个选项，请使用此

<div class="MuiPaper-root MuiMenu-paper MuiPaper-elevation8 MuiPopover-paper MuiPaper-rounded" role="document" tabindex="-1" style="opacity: 1; transform: none; min-width: 491px; transition: opacity 381ms cubic-bezier(0.4,0.2,1) 0ms,transform 254ms cubic-bezier(0.4,1) 0ms; top: 80px; left: 16px; transform-origin: -1px 478.513px;">
<ul class="MuiList-root MuiMenu-list MuiList-padding" role="listbox" tabindex="-1" style="padding-right: 17px; width: calc(100% + 17px);">
<li class="MuiButtonBase-root MuiListItem-root MuiMenuItem-root MuiMenuItem-gutters MuiListItem-gutters MuiListItem-button" tabindex="-1" role="option" aria-disabled="false" variant="outlined" data-value="testOne">Test One<span class="MuiTouchRipple-root"></span></li>
<li class="MuiButtonBase-root MuiListItem-root MuiMenuItem-root MuiMenuItem-gutters MuiListItem-gutters MuiListItem-button" tabindex="-1" role="option" aria-disabled="false" variant="outlined" data-value="testTwo">Test Two<span class="MuiTouchRipple-root"></span></li>

如果要在选择框上单击后删除第一个选项，请使用

$(document).ready(function()
    {
       $("#q_c_0_a_0_name").children().first().remove();           
    });