Type 'string' is not assignable to type '"menu" | "selectedMenu" | undefined'

Because variant only receive type is "menu" | "selectedMenu" | undefined. But you pass a value is "menu" without type. So it will auto has type is string. You need to define a type for variant. Like this:

const MenuProps = {
  variant: 'menu' as 'menu',
  ...
};

You can import the MenuProps type from Material-UI with:

import { MenuProps as MenuPropsType } from "@material-ui/core/Menu";

Then you can use this like the following:

const MenuProps: Partial<MenuPropsType> = {
  variant: "menu",
  PaperProps: {
    style: {
      maxHeight: ITEM_HEIGHT * 4.5 + ITEM_PADDING_TOP,
      width: 250
    }
  }
};

You use Partial<MenuPropsType> (as it also is in the Select type definition) because Select automatically adds the required open prop (if you remove Partial, it will complain about the open prop not being specified).

Here's a working example based on your code:

import React from "react";
import { MenuProps as MenuPropsType } from "@material-ui/core/Menu";
import Input from "@material-ui/core/Input";
import MenuItem from "@material-ui/core/MenuItem";
import Select from "@material-ui/core/Select";

const ITEM_HEIGHT = 48;
const ITEM_PADDING_TOP = 8;
const MenuProps: Partial<MenuPropsType> = {
  variant: "menu",
  PaperProps: {
    style: {
      maxHeight: ITEM_HEIGHT * 4.5 + ITEM_PADDING_TOP,
      width: 250
    }
  }
};

const names = [
  "Oliver Hansen",
  "Van Henry",
  "April Tucker",
  "Ralph Hubbard",
  "Omar Alexander",
  "Carlos Abbott",
  "Miriam Wagner",
  "Bradley Wilkerson",
  "Virginia Andrews",
  "Kelly Snyder"
];

export default function MultipleSelect() {
  const [personName, setPersonName] = React.useState<string[]>([]);

  const handleChange = (event: React.ChangeEvent<{ value: unknown }>) => {
    setPersonName(event.target.value as string[]);
  };

  return (
    <Select
      multiple
      value={personName}
      onChange={handleChange}
      input={<Input />}
      MenuProps={MenuProps}
    >
      {names.map((name) => (
        <MenuItem key={name} value={name}>
          {name}
        </MenuItem>
      ))}
    </Select>
  );
}